Bifurcations of nontrivial solutions of a cubic Helmholtz system

This paper presents local and global bifurcation results for radially symmetric solutions of the cubic Helmholtz system \begin{equation*} \begin{cases} -\Delta u - \mu u = \left( u^2 + b \: v^2 \right) u&\text{ on } \mathbb{R}^3, \\ -\Delta v - \nu v = \left( v^2 + b \: u^2 \right) v&\text{ on } \mathbb{R}^3. \end{cases} \end{equation*} It is shown that every point along any given branch of radial semitrivial solutions $(u_0, 0, b)$ or diagonal solutions $(u_b, u_b, b)$ (for $\mu = \nu$) is a bifurcation point. Our analysis is based on a detailed investigation of the oscillatory behavior of solutions at infinity that are shown to decay like $\frac{1}{|x|}$ as $|x|\to\infty$.


Introduction and main results
Systems of two coupled nonlinear Helmholtz equations arise, for instance, in models of nonlinear optics. In this paper, we analyze the physically relevant and technically easiest case of a Kerr-type nonlinearity in N = space dimensions, that is, we study the system for given µ, ν > and a constant coupling parameter b ∈ R. We are mostly interested in existence results for fully nontrivial radially symmetric solutions of this system that we will obtain using bifurcation theory. Such an approach is new in the context of nonlinear Helmholtz equations or systems. In order to describe the methods used in related works we brie y discuss the available results for scalar nonlinear Helmholtz equations of the form Here, the main di culty is that solutions typically oscillate and do not belong to H (R N ). In the past years, Evéquoz and Weth have developed several methods allowing to nd nontrivial solutions of (1) under certain conditions on Q and p, some of which we wish to mention. In [1,2], they discuss the case of compactly supported Q and < p < * := N N− . The idea in [1] is to solve an exterior problem where the nonlinearity vanishes and knowledge about the far-eld expansion of solutions is available. The remaining problem on a bounded domain can be solved using variational techniques. The approach in [2] uses Leray-Schauder continuation with respect to the parameter λ in order to nd solutions of (1). Existence of solutions under the assumption that Q ∈ L ∞ (R N ) decays as |x| → ∞ or is periodic is proved in [3] using a dual variational approach, which yields (dual) ground state solutions and, in the case of decaying Q, in nitely many bound states. The technique relies on the Limiting Absorption Principle of Gutiérrez, see Theorem 6 in [4], which leads to the additional constraint (N+ ) N− < p < * . Furthermore, assuming that Q is radial, the existence of a continuum of radially symmetric solutions of (1) has been shown by Montefusco, Pellacci and the rst author in [5], generalizing earlier results in [1]. Their results rely on ODE techniques and only require p > and a monotonicity assumption on Q.
To our knowledge, the only available result on nonlinear Helmholtz systems like (H) has been provided by the authors in [6] where, using the methods developed in [3], the existence of a nontrivial dual ground state solution is proved for the system for N ≥ , Z N -periodic coe cients a, b ∈ L ∞ (R N ) with a(x) ≥ a > , ≤ b(x) ≤ p − and (N+ ) N− < p < * . Under additional easily veri able assumptions the ground state can be shown to be fully nontrivial, i.e., both components are nontrivial. Assuming constant coe cients and working on spaces of radially symmetric functions, this variational existence result for dual ground states extends to the case p = , N = which we dicuss in the present paper. In contrast to [6] we construct fully nontrivial radial solutions for arbitrarily large and small b ∈ R that, however, need not be dual ground states.
Motivated by the decay properties of radial solutions of nonlinear Helmholtz equations in [5], e.g. Theorem 1.2 (iii), we look for solutions of (H) in the Banach space X where, for q ≥ , Xq := w ∈ C rad (R , R) | w Xq < ∞ with w Xq := sup x∈R ( + |x| ) q |w(x)|. ( Working on these spaces, we will be able to derive compactness properties which are crucial when proving our bifurcation results. Throughout, we discuss classical, radially symmetric solutions u, v ∈ X ∩ C (R ) of the system (H) and related equations. Let us remark here only brie y that, using elliptic regularity, all weak solutions u, v ∈ L rad (R ) are actually smooth and, thanks to Proposition 6 in the next section, belong to X ∩ C (R ).
We study bifurcation of solutions (u, v, b) of the nonlinear Helmholtz system (H) from a branch of semitrivial solutions of the form Here u : R → R denotes any of the uncountably many nontrivial radial solutions of the scalar Helmholtz equation which all belong to the space X , see [5]. In contrast to the Schrödinger case, we will demonstrate that every point in Tu is a bifurcation point for fully nontrivial solutions of (H). Our strategy is to use bifurcation from simple eigenvalues with b acting as a bifurcation parameter. The existence of isolated and algebraically simple eigenvalues will be ensured by assuming radial symmetry and by imposing suitable conditions on the asymptotic behavior of the solutions u, v. For τ, ω ∈ [ , π), we de ne S ⊆ X × X × R \ Tu as the set of all solutions (u, v, b) ∈ X × X × R \ Tu of (H) satisfying the asymptotic conditions for some c , c ∈ R. Propositions 4 and 6 below show that it is natural to assume such an asymptotic behavior for solutions of (H). For notational convenience, we do not denote the dependence of S and of the asymptotic conditions (A) on the choice τ, ω ∈ [ , π). As we will show in Proposition 6, there exists a unique τ = τ (u ) ∈ [ , π) such that, for τ ∈ [ , π), With that, we obtain the following Theorem 1. Let µ, ν > , x any u ∈ X \ { } solving the nonlinear Helmholtz equation (h) and choose τ ∈ [ , π) \ {τ } according to (N). Then, for every ω ∈ [ , π), there exists a strictly increasing sequence ) has a neighborhood where C k is a smooth curve in X × X × R which, except for the bifurcation point, consists of fully nontrivial solutions.
The main tools in proving this statement are the Crandall-Rabinowitz Bifurcation Theorem, which will be used to show the local statement (ii) of Theorem 1, and Rabinowitz' Global Bifurcation Theorem, which will provide (i). For a reference, see [7], Theorem 1.7 and [8], Theorem 1.3. We add some remarks the proof of which will also be given in Section 3.

Remark 2. (a)
We will also see that fully nontrivial solutions of (H) satisfying the asymptotic condition (A) bifurcate from some point (u , , b) ∈ Tu if and only if b = b k (ω) for some k ∈ Z. Moreover, the proof will show that the values b k (ω) do not depend on the choice of τ.
where ≤ ω < π, k ∈ Z is strictly increasing and onto with b k (ω) → ±∞ as k → ±∞. In particular, every point (u , , b) ∈ Tu , b ∈ R, is a bifurcation point for fully nontrivial radial solutions of (H), which is in contrast to Schrödinger systems where bifurcation points are isolated, cf. [9], Satz 2.1.6. (c) Close to the respective bifurcation point (u , , b k (ω)) ∈ Tu , each continuum C k is characterized by a phase parameter ων(v + b u ) = ω + kπ derived from the asymptotic behavior of v (see (10)). It seems that, in the Helmholtz case of oscillating solutions, the integer k takes the role of the nodal characterizations in the Schrödinger case, cf. Satz 2.1.6 in [9]. That phase parameter is constant on connected subsets of the continuum until it possibly runs into another family of semitrivial solutions Tu with u ≠ u ; unfortunately we cannot provide criteria deciding whether or not this happens. For this reason we cannot claim that C k contains an unbounded sequence of fully nontrivial solutions. [5]. Then w := d dδ δ= u δ satis es −∆w − µw = u w on R , w( ) = . We de ne τ ∈ [ , π) as the constant appearing in the asymptotic expansion of w, for some unique c ≠ and τ ∈ [ , π), see Proposition 6. With that in mind, the condition τ ≠ τ is a nondegeneracy condition which by means of (N) ensures that the simplicity requirements of the abovementioned bifurcation theorems are satis ed.
Our results are inspired by known bifurcation results for the nonlinear Schrödinger system where one assumes λ , λ > in contrast to (H). We focus on bifurcation results by Bartsch, Wang and Wei in [10] and Bartsch, Dancer and Wang in [11] and refer to the respective introductory sections for a general overview of methods and results for (3). In Theorem 1.1 of [10] the authors show that a continuum consisting of positive radially symmetric solutions (u, v, λ , λ , µ , µ , b) of (3) with topological dimension at least bifurcates from a two-dimensional set of semipositive solutions (u, v) = (u λ ,µ , ) parametrized by λ , µ > . The existence of countably many bifurcation points giving rise to sign-changing radially symmetric solutions was proved by the rst author in his dissertation thesis (Satz 2.1.6 of [9]). In Theorem 1 above, we analyze the corresponding case of bifurcation from a semitrivial family Tu in the Helmholtz case. In contrast to the Schrödinger case, our result shows bifurcation at every point in the topology of X × X ×R, see Remark 2 (b). Looking more closely, we nd the same structure of discrete bifurcation points as in the Schrödinger case when xing parameters τ, ω prescribing the oscillatory behavior of solutions as |x| → ∞ as in the condition (A). In the Schrödinger case, the bifurcating solutions are characterized by their nodal structure; in the Helmholtz case, we use instead a condition on the "asymptotic phase" of the solution (disguised as an integral), which at least close to the j-th bifurcation point takes the value ω + jπ as described in Remark 2 (c).
Similar observations can be made for bifurcation from families of diagonal solutions of the Schrödinger system (3) in the special case N = , and λ = λ > and µ , µ > ; in order to keep the presentation short, we assume in addition µ = µ = . Bartsch, Dancer and Wang proved in [11] the existence of countably many mutually disjoint global continua of solutions bifurcating from some diagonal solution family of the form is a nondegenerate solution of −∆u + u = u . Moreover, having introduced a suitable labeling of the continua, the authors showed that the k-th continuum consists of solutions where the radial pro le of u − v has exactly k − nodes, cf. Theorem 2.3 in [11].
We provide a counterpart for the Helmholtz system (H) in our second result, Theorem 3, using the same functional analytical setup as in Theorem 1. Here we assume ν = µ. For nonzero u solving (h), we can then introduce the diagonal solution family Given τ, ω ∈ [ , π), we denote by S the set of all solutions (u, v, b) ∈ X × X × R \ Tu of the nonlinear Helmholtz system (H) with for some c , c ∈ R. Our existence result for fully nontrivial solutions of (H) bifurcating from Tu with asymptotics (A diag ) reads as follows.
) has a neighborhood where the set C k contains a smooth curve in X × X × R which, except for the bifurcation point, consists of fully nontrivial, non-diagonal solutions.
Again, similar statements as in Remark 2 can be proved. In particular, one can check that every point on Tu is a bifurcating point by a suitable choice of ω.
We point out that our methods in Theorems 1 and 3 also apply for nontrivial radial solutions of −∆u − µu = −u on R and corresponding modi cations in the system (H). Such solutions u exist in the Helmholtz case (but not in the Schrödinger case) and belong to the space X , see Theorem 1.2 in [5].
Let us give a short outline of this paper. In Section 2, we introduce the concepts and technical results we use in the proof of Theorems 1 and 3, which are presented in Section 3 and Section 4. In the nal section, we provide the proofs of the auxiliary results of Section 2.

Properties of the scalar problem
The main challenge in proving Theorem 1 is the thorough analysis of the linearized problem provided in this chapter. Throughout, we x λ > and discuss the linear Helmholtz equation for some f ∈ X , where X is de ned in (2). We will frequently identify radially symmetric functions x → w(x) with their pro les; in particular, we denote by w := ∂r w, w = ∂ r w the radial derivatives. The results we establish in this section will demonstrate how to rewrite the system (H) in a way suitable for Bifurcation Theory.

. Representation Formulas
First, we discuss a representation formula for solutions of the linear Helmholtz equation (4). The results resemble a more general Representation Theorem by Agmon, Theorem 4.3 in [12]. We introduce the fundamental solutions of the equation −∆w − λw = on R . We observe thatΨ λ is, up to multiplication with a constant, its unique global classical solution. We will frequently require knowledge of the mapping properties of convolutions with Ψ λ resp.Ψ λ . Various results of such type have been found by Evéquoz and Weth in [3] and further publications, assuming f ∈ L p (R N ), w ∈ L p (R N ) for suitable p, p ∈ ( , ∞). In the spaces X resp. X , which satisfy the continuous embeddings we prove the following statements.
(d) for f ∈ X , the pro le of w := R λ f and its radial derivative satisfy as r → ∞ As a consequence of Proposition 4, we state the representation formulae we require later to construct the functional analytic setting in the proof of Theorem 1. For ω ∈ ( , π), we de ne which provide solutions of the Helmholtz equation (4) the asymptotic behavior of which is described by the phase parameter ω as follows.

Corollary 5.
Let ω ∈ ( , π) and f ∈ X . Then, for w ∈ X , we have We point out that the operator R ω λ is not well-de ned for ω = due to the pole of the cotangent. We will comment on suitable modi cations during the proofs of Theorems 1 and 3.

. The Asymptotic Phase
Frequently, equations of interest will take the form (4) with f = g · w for some g ∈ X , see (2). We can then use ODE methods, more speci cally the Prüfer transformation, to discuss the corresponding initial value problem for the pro les, Proposition 6. Assume g ∈ X . Then the initial value problem (9) has a unique (global) solution w : [ , ∞) → R which satis es as r → ∞ for some ϱ λ (g) > and ω λ (g) ∈ R. Here, the value of ω λ (g) is given by In particular, given u ∈ X \ { } solving (h), Proposition 6 with g := u ∈ X shows that the nondegeneracy condition (N) holds with τ ∈ [ , π) such that ωµ( u ) ∈ τ + πZ.
Comparing Proposition 6 with Corollary 5, we observe that Proposition 6 guarantees ϱ λ (g) > , that is, the solution has a nonvanishing term of leading order as r → ∞. The asymptotic conditions imposed in Corollary 5 with f = g · w now take the form ω λ (g) ∈ ω + πZ. Such boundary conditions at in nity will provide operators with spectral properties suitable for building the functional analytic framework in which to prove Theorem 1. As a rst auxiliary result, we prove the following continuity property.
When studying eigenvalue problems of a linearization of (H), we need to know the dependence of the asymptotic phase ω λ (b u ) on the parameter b ∈ R.

. The spectrum of the linearization
In the proof of Theorem 1, we will rewrite the nonlinear Helmholtz system (H) in the form for some τ, ω ∈ ( , π), which additionally imposes a certain asymptotic behavior on the solutions, see Corollary 5. In order to analyze the linearized problem, we x some nontrivial u ∈ X ∩C (R ) with −∆u −µu = u on R and study the spectra of the linear operators which are compact thanks to Proposition 4 (b).

Proposition 9.
Let ω ∈ ( , π), λ > and u as before. Then the spectrum of R ω λ is Moreover, all eigenvalues are algebraically simple, and the sequence (b k (ω, λ, u )) k∈Z is strictly increasing and unbounded below and above.
This excludes the case ω = , even though the values b k ( , λ, u ) ∈ R, k ∈ Z, can be de ned accordingly. Indeed, the rst step of the proof of Proposition 9 above shows for all ω ∈ [ , π):
We de ne the map with the convolution operators R τ µ , R ω ν : X → X from (8). Observe that F is well-de ned since u, v, w ∈ X implies uvw ∈ X . Recalling Corollary 5 and (h), we have So we aim to nd nontrivial zeros of F. Second, we observe that F has a trivial solution family, that is is a compact perturbation of the identity on X × X since the operators R τ µ , R ω ν : X → X are compact thanks to Proposition 4 (b). Moreover, F is twice continuously Fréchet di erentiable; we have for φ, ψ ∈ X and b ∈ R, denoting by D the Fréchet derivative w.r.t. the w and v components, with compact linear operators R τ µ , R ω ν : X → X as in (11). We deduce that, due to (N) and τ ≠ τ , We show in the following that it is also su cient.
We apply the Crandall-Rabinowitz Theorem at the point ( , , b k (ω)). As F( · , b) is a compact perturbation of the identity on X × X , the Riesz-Schauder Theorem implies that DF( , , b k (ω)) is a Fredholm operator of index zero with one-dimensional kernel spanned by ( , ψ k ), see above. To verify the transversality condition, we rst compute which contradicts the algebraic simplicity of the eigenvalue b k (ω) − of R ω ν proved in Proposition 9. Thus ∂ b DF( , , b k (ω))[( , ψ k )] ∉ ran DF( , , b k (ω)), and the Crandall-Rabinowitz Theorem provides the smooth curve of solutions of F(w, v, b) = as in (ii). Further, possibly shrinking the neighborhood where the local result holds, we may w.l.o.g. assume fully nontrivial solutions (u + w, v) of (H) since the direction of bifurcation is given by ( , ψ k ).
We have already seen that F ( · , b), b ∈ R, is a compact perturbation of the identity on X × X . Thus the application of Rabinowitz' Global Bifurcation Theorem only requires to verify that the index of F( · , b) in ( , ) changes sign at each value b = b k (ω), k ∈ Z. By the identity (12), for b ∉ {b k (ω) | k ∈ Z}, ind X ×X F( · , b), ( , ) = ind X ×X DF ( , , b), ( , ) (12) = ind X I − R τ µ , · ind X I − b R ω ν , , and hence ind X ×X F( · , b), ( , ) changes sign at b = b k (ω) if and only if so does ind X I − b R ω ν , . The latter change of index occurs since b k (ω) is an isolated eigenvalue of algebraic multiplicity 1 of R ω ν , see Proposition 9.
The Global Bifurcation Theorem by Rabinowitz asserts that (u , , b k (ω)) ∈ S and that the associated connected component C k of S is unbounded or returns to Tu at some point (u , , b * ). We prove that, in any case, the component is unbounded.
The asymptotic phase satis es ων(b k (ω)u ) = ω+kπ by de nition of b k (ω), see Step 1, and ων(v +bu ) ∈ ω+πZ for all (u, v, b) ∈ C k with v ≠ . This is due to (A) and Proposition 6. So if all elements (u, v, b) ∈ C k \Tu satisfy v ≠ , then as a consequence of the continuity of ων (see Proposition 7) and of the fact that C k is connected, we infer that ων(v + bu ) = ω + kπ for all (u, v, b) ∈ C k . Let us now assume that C k returns to the trivial family in some point (u , , b * ) ∈ Tu , b * ≠ b k (ω). Then ων(b * u ) ≠ ω + kπ by strict monotonicity (see The case ω = and τ ∈ ( , π) \ {τ }.
We recall that, in case ω = , the map F resp. R ω ν is not well-de ned due to the pole of the cotangent. To write down a suitable replacement, we use the Hahn-Banach Theorem to de ne functionals α (ν) , β (ν) ∈ X with the following property: For w ∈ X with for some αw , βw ∈ R, we have α (ν) (w) := αw and β (ν) (w) := βw. We then de ne for σ = ± Using Step 2: Local Bifurcation.
Integrating by parts and exploiting the asymptotic behavior of ψ resp. ψ k and their derivatives, see Proposition 4 (d) and equation (13), this nally implies which is a contradiction as R → ∞; hence transversality holds.
We apply Rabinowitz' Global Bifurcation Theorem from [13], Theorem II.3.3, which as above yields unbounded connected components C k ⊆ S once we show that the index More precisely, we analyze bifurcation at b k ( ) ≥ using the map G+ and at b k ( ) < using G−. In the following, we present the main ideas how to verify that is an algebraically simple eigenvalue of K b k ( ) and that the corresponding perturbed eigenvalue λ b ≈ of K b for b ≈ b k ( ) crosses as b crosses b k ( ). For the existence, algebraic simplicity and continuous dependence of λ b on b we refer to Kielhöfer's book [13], p. 203.

Perturbation of the eigenvalue.
Throughout the following lines, we consider a perturbed value b ≈ b k ( ), b ≠ b k ( ) and the corresponding eigenpair with K b w b = λ b w b . The latter implies and hence λ b ≠ due to β (ν) (w b ) ≠ , see Proposition 8. We recall that where the second identity can be deduced comparing the expansions in equation (13) and in Corollary 5 resp. Proposition 6. We now discuss the values b k ( ) ≥ , i.e. σ = + . In case b > b k ( ) we show that λ b > . Assuming λ b < , we infer from (16) that sgn α (ν) (w b ) ≠ sgn β (ν) (w b ) and thus ων(bλ − b u ) ∈ − π , + πZ due to (17). But since bλ − b > b k ( ), the monotonicity stated in Proposition 8 implies ων(bλ − b u ) ∈ ων(b k ( )u )+ , π ⊆ , π + πZ, a contradiction. In the same way, for b < b k ( ), we can show that λ b < . Following the same strategy, we see for We have thus proved that, as b crosses b k ( ), the perturbed eigenvalue λ b crosses λ b k ( ) = and hence the sign of the Leray-Schauder index ind X ×X Gσ( · , b), ( , ) changes at b = b k ( ) for all k ∈ Z and for σ ∈ {± } chosen as above.

The case τ = .
This is covered by rede ning the rst components of F resp. Gσ,  ) for ω ∈ [ , π), k ∈ Z, we infer strict monotonicity and surjectivity of the map In Steps 2 we have seen that in a neighborhood of the bifurcation point (u , , b k (ω)), the continuum C k contains only fully nontrivial solutions apart from (u , , b k (ω)) itself. In Step 3, we infer for all (u, v, b) ∈ C k from this neighborhood that the asymptotic phase of v satis es ων(v + bu ) = ω + kπ. More generally, ων(v + bu ) = ω + kπ holds on every connected subset of C k containing (u , , b k (ω)) but no other semitrivial solution with v = . (d) By Proposition 6, the (formally derived) initial value problem has a unique radial solution with c = ϱµ( u ) ≠ and τ = ωµ( u ) ∈ [ , π).

Proof of Theorem 3
We now prove the occurence of bifurcations from the diagonal solution family Tu . To this end we rst rewrite the system (H) in an equivalent but more convenient way. Looking for solutions (u, v, b) ∈ X × X × R \ Tu , we introduce the functions w , w ∈ X via A few computations then yield that bifurcation at the point (u b , u b , b) occurs if and only if we have bifurcation from the trivial solution of the nonlinear Helmholtz system and the asymptotic conditions (A diag ) are equivalent to as |x| → ∞ for some c , c ∈ R. As in the proof of Theorem 1, the functional analytical setting in the special cases ω = or τ = is di erent from the general one since a substitute for the operators R τ µ , R ω µ has to be found, see the de nition of Gσ in the proof of Theorem 1. In order to keep the presentation short we only discuss the case τ, ω ∈ ( , π) and refer to the proof of Theorem 1 for the modi cations in the remaining cases. So we introduce the map F : ) is a compact perturbation of the identity on X × X and it remains to nd bifurcation points for F(w , w , b) = . First we identify candidates for bifurcation points, i.e. those b ∈ (− , ∞) where ker DF ( , , b) is nontrivial. Using we get that nontrivial kernels occur exactly if −b +b = b k (ω) for some k ∈ Z, cf. Steps 1 in the previous proof. For the analogous result in the Schrödinger case, see Lemma 3.1 [11]. So Using the algebraic simplicity of ψ k proved in Proposition 9 we infer exactly as in the proof of Theorem 1 that the transversality condition holds and that the Leray-Schauder index changes at the bifurcation point. So, choosing b k (ω) := −b k (ω) +b k (ω) for all k ∈ Z with b k (ω) > − , the Crandall-Rabinowitz Theorem and Rabinowitz' Global Bifurcation Theorem yield statements (ii) and (i) of the Theorem, respectively. We remark that, to be consistent with the labeling in the Theorem, we might have to shift the index in such way that b (ω) ≤ − < b (ω).
Unboundedness of the components can also be deduced as before. Indeed, assuming that C k is bounded, it returns to Tu at some point (u b * , u b * , b * ) ≠ (u b k (ω) , u b k (ω) , b k (ω)) by Rabinowitz' Theorem. We then infer that the phase ωµ(( + b)w + ( − b)(w + u b ) ) cannot be constant along C k . Due to Proposition 6 applied to w in (18), this requires the existence of some element (u, v, b) ∈ C k with w = (v − u) = , and hence the associated unbounded diagonal family belongs to C k .

Proofs of the Results in Section 2
Before proving Proposition 4, we state two auxiliary results. The rst one provides a formula for the Fourier transform of radially symmetric functions, see e.g. [14], p. 430.
Lemma 12. For f ∈ X and x ∈ R \ { }, we have

. Proof of Proposition 4
We now prove, one by one, the assertions of Proposition 4 for convolutions with Φ λ in place of αΨ λ +αΨ λ . The latter (real-valued) case can be deduced from the former via Φ λ = Ψ λ + iΨ λ . Unless stated otherwise, we extend norms de ned on spaces of real-valued functions to complex-valued functions g : R → C by considering the respective norm of |g| : R → R. (a) is a consequence of Theorem 2.1 in [3]. The solution properties stated in (c) can be veri ed by direct computation.
Due to the continuous embedding X ⊆ L rad (R ), see (6), the convolution is well-de ned for f ∈ X . Using Young's convolution inequality, we get ≤ C · f X for some C ≥ . Next, by means of Lemma 12, we estimate for Step 2: Proof of (d). Asymptotics of w and w .
Given f ∈ X , we let w := Φ λ * f . Then for r = |x| > , Lemma 12 implies To understand the asymptotic behavior of the radial derivative w , a short calculation shows that the auxiliary function δ(r) Then for r > , we nd τr ∈ ( , ) with δ(r + ) = δ(r) + δ (r) + δ (r + τr), whence also δ (r) = O r . This shows the asserted properties of w since As a consequence of equation (20), we derive the formula stated forΨ λ * f . Due to (c),Ψ λ * f is a radial solution of the homogeneous Helmholtz equation −∆w − λw = on R and hence a scalar multiple ofΨ λ itself. The asymptotics in (20) justify the asserted constant.
We consider a bounded sequence (fn)n in the space X and aim to prove convergence of a subsequence of (un)n, un := Φ λ * fn, in the space X . First, due to the continuous embeddings into re exive L p spaces stated in (6), we can pass to a subsequence with fn k f weakly in L (R ) ∩ L (R ), un k u weakly in L (R ) for some f ∈ L rad (R ) ∩ L rad (R ), u ∈ L rad (R ). Then the regularity properties in Proposition A.1 in [3] and the Rellich-Kondrachov Embedding Theorem 6.3 in [15] allow to extract a subsequence with un k → u strongly in C loc (R ), in particular for any R > On the other hand, on the unbounded set R \B R ( ), convergence in X follows essentially from the asymptotic expansion in equation (20) where w.l.o.g.fn k ( Thus given ε > , we can choose R(ε) > large enough and k(ε) ∈ N such that (21) and (22) imply un k − un l X < ε for all k, l ≥ k(ε). Hence (un k ) k∈N is a Cauchy sequence in X .

. Proof of Proposition 7
We consider gn , g in X with gn → g in X and aim to show that ω λ (gn) → ω λ (g ). By ϕn ∈ C (( , ∞)) ∩ C([ , ∞)) we denote the unique solution of Then we have pointwise convergence, ϕn(r) → ϕ (r) for all r ≥ . Indeed, let us x any R > and estimate for ≤ r ≤ R and n ∈ N Thus, by Gronwall's Lemma, we have for ≤ r ≤ R Since gn → g in X , we conclude ϕn → ϕ locally uniformly on [ , ∞), in particular pointwise. Now we can deduce the convergence of the asymptotic phase, which follows by dominated convergence since sup n∈N gn X < ∞.

. Proof of Proposition 8
Let us rst recall that, given the assumptions of Proposition 8, equation (10) implies for b ∈ R We immediately see that ω λ ( ) = and sgn ω λ (b u ) = sgn (b) for all b ∈ R \ { }. Further, continuity of b → ω λ (b u ) is a consequence of Proposition 7. The assertions are proved once we show that b → ω λ (b u ) is strictly increasing with ω λ (b u ) → ±∞ as b → ±∞.
Step 1: Strict monotonicity. We let b < b , de ne and observe that χ is bounded with ≤ |χ(r)| ≤ and continuous. ψ := ϕ b − ϕ b satis es The unique solution is given by the Variation of Constants formula. We have since the integrand is nonnegative and not identically zero.
By the uniqueness statement of the Picard-Lindelöf Theorem, u ≢ requires u ( ) ≠ . We can thus choose r > with To keep notation short, we let in this paragraph ξ := λ u ( ). We have for b > We now study the modi ed initial value problem For ≤ r ≤ r with r ∉ π + πZ, its unique solution is given by the expression sin (t) dt and hence ϕ b (r ) → ∞ implies ω λ (b u ) → ∞ as b → ∞.
For b < − , we introduce which is well-de ned due to − |b| We prove below that r b → ∞ as b → −∞. Then for r ≥ r b , equation (29) and ϕ b ≤ imply Then the asymptotic phase satis es It remains to prove that r b → ∞ as b → −∞. We assume for contradiction that we nd a subsequence (b k ) k∈N andr > with b k ↘ −∞, r b k →r as k → ∞. Then, since ϕ b k ≤ and due to equation (29), we have for su ciently large k ∈ N We conclude sin(ϕ b k (r) √ λ) ≥ |b k | − and hence as k → ∞ → −∞ since u > almost everywhere and r b k →r. On the other hand, for every k ∈ N, the di erential equation ϕ = + b k λ u (r) sin (ϕ √ λ) states that ϕ b k (r) = implies ϕ b k (r) = . Thus ϕ b k cannot attain negative values, which contradicts the limit calculated before.
We nd the eigenfunctions of R ω λ , that is, we look for such η ∈ R, η ≠ and nontrivial w ∈ X that R ω λ w = η · w. Corollary 5 implies that this is equivalent to η ∈ R, η ≠ and nontrivial w ∈ X ∩ C (R ), for some c ∈ R. By Proposition 6, such an eigenfunction exists if and only if ω λ η u = ω + kπ for some k ∈ Z; in this case, c ≠ and every eigenspace is one-dimensional since the radially symmetric solution w is unique up to multiplication by a constant. Since we have seen in Proposition 8 that R → R, b → ω λ (b u ) is strictly increasing and onto, we can de ne b k (ω, λ, u ) via ω λ (b k (ω, λ, u ) u ) = ω + kπ for all k ∈ Z, and conclude Step 2: Simplicity.
It remains to show that the eigenvalues are algebraically simple. We consider an eigenvalue η := b k (ω,λ,u ) of R ω λ with eigenspace ker R ω λ − ηI X = span {w}. We have to prove that ker R ω λ − ηI X = ker R ω λ − ηI X .
for some c , c ∈ R. Let us de ne q(r) = r (w (r)v(r) − v (r)w(r)) for r ≥ . Then, using (31), we nd q (r) = η r u (r) · w (r) for r ≥ . Hence q is monotone on [ , ∞) with q( ) = . On the other hand, the asymptotic expansions in (32) imply q(r) = O r as r → ∞. We conclude q(r) = for r ≥ . Since all zeros of w are simple, one can deduce that v(r) = c · w(r) for all r ≥ and some c ∈ R, and thus v ∈ ker R ω λ − ηI X , a contradiction.